Accepted Papers
Published Journal Articles
■ Free boundary problems in fluid dynamics equations
♣ MHD equations
- Hao, Chengchun and Yang, Siqi;
Splash singularity for the free boundary incompressible viscous MHD
Journal of Differential Equations, 379: 26–103, 2024. 
In this paper, we prove the existence of smooth initial data for the two-dimensional free
boundary incompressible viscous magnetohydrodynamics (MHD) equations, for which
the interface remains regular but collapses into a splash singularity (self-intersects in at
least one point) in finite time. The existence of the splash singularities is guaranteed by
a local existence theorem, in which we need suitable spaces for the modified magnetic
field together with the modification of the velocity and pressure such that the modified
initial velocity is zero, and a stability result which allows us to construct a class of
initial velocities and domains for an arbitrary initial magnetic field. It turns out that
the presence of the magnetic field does not prevent the viscous fluid from forming splash
singularities for certain smooth initial data.
@article{HY24spvMHD,
author ={Hao, Chengchun and Yang, Siqi},
journal={Journal of Differential Equations},
title ={Splash singularity for the free boundary incompressible viscous MHD},
year ={2024},
volume ={379},
number ={},
pages ={26–103},
doi={10.1016/j.jde.2023.10.001},
}
- Fu, Jie and Hao, Chengchun and Yang, Siqi and Zhang, Wei;
A Beale-Kato-Majda criterion for free boundary incompressible ideal magnetohydrodynamics
Journal of Mathematical Physics, 64(9): No.091505, 16pp, 2023.
We prove a continuation criterion for the free boundary problem of three-dimensional incompressible ideal magnetohydrodynamic (MHD) equations in a bounded domain, which is analogous to the theorem given in [Beale, Kato and Majda 1984 Comm. Math. Phys. 94 61--66]. We combine the energy estimates of our previous works [Hao and Luo 2014 {Arch. Ration. Mech. Anal. 212 805--847] on incompressible ideal MHD and some analogous estimates in [Ginsberg 2021 SIAM J. Math. Anal.53(3) 3366--3384] to show that the solution can be continued as long as the curls of the magnetic field and velocity, the second fundamental form and injectivity radius of the free boundary and some norms of the pressure remain bounded, provided that the Taylor-type sign condition holds.
@article{FHYZ23a,
author ={Fu, Jie and Hao, Chengchun and Yang, Siqi and Zhang, Wei},
journal={Journal of Mathematical Physics},
title ={A Beale-Kato-Majda criterion for free boundary incompressible ideal magnetohydrodynamics},
year ={2023},
volume ={64},
number ={9},
pages ={No.091505, 16pp},
doi={10.1063/5.0167954},
}
- Hao, Chengchun and Zhang, Wei;
A priori estimates for free boundary problem of 3D incompressible inviscid rotating Boussinesq equations
Zeitschrift für angewandte Mathematik und Physik, 74(2): No.80, 21pp, 2023.
In this paper, we consider the three-dimensional rotating Boussinesq equations (the “primitive” equations of geophysical fluid flows). Inspired by Christodoulou and Lindblad (Commu. Pure Appl. Math. 53:1536–1602, 2000), we establish a priori estimates of Sobolev norms for free boundary problem of inviscid rotating Boussinesq equations under the Taylor-type sign condition on the initial free boundary. Using the same method, we can also obtain a priori estimates for the incompressible inviscid rotating MHD system with damping.
@article{HZ23b,
author ={Hao, Chengchun and Zhang, Wei},
journal={Zeitschrift für angewandte Mathematik und Physik},
title ={A priori estimates for free boundary problem of 3D incompressible inviscid rotating Boussinesq equations},
year ={2023},
volume ={74},
number ={2},
pages ={No.80, 21pp},
doi={10.1007/s00033-023-01974-2},
}
- Hao, Chengchun and Luo, Tao;
Some results on free boundary problems of incompressible ideal magnetohydrodynamics equations
Electronic Research Archive, 30(2): 404-424, 2022.
We survey some recent results related to free boundary problems of incompressible ideal magnetohydrodynamics equations, and present the main ideas in the proofs of the ill-posedness in 2D when the Taylor sign condition is violated given [1], and the well-posedness of a linearized problem given in [2] in general n-dimensions $(n\geqslant 2)$ when the Taylor sign condition is satisfied and the free boundaries are closed.
@article{HL22ERA,
author ={Hao, Chengchun and Luo, Tao},
journal={Electronic Research Archive},
title ={Some results on free boundary problems of incompressible ideal magnetohydrodynamics equations},
year ={2022},
volume ={30},
number ={2},
pages ={404-424},
doi={10.3934/era.2022021},
}
- Hao, Chengchun and Luo, Tao;
Well-posedness for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations
Journal of Differential Equations, 299: 542-601, 2021.
The well-posedness theory is studied for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations in a bounded domain. We express the magnetic field in terms of the velocity field and the deformation tensors in Lagrangian coordinates, and substitute it into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, the velocity equation is linearized with respect to the position vector field whose time derivative is the velocity. In this formulation, a key idea is to use the Lie derivative of the magnetic field taking the advantage that the magnetic field is tangential to the free boundary and divergence free. This paper contributes to the program of developing geometric approaches to study the well-posedness of free boundary problems of ideal magnetohydrodynamics equations under the condition of Taylor sign type for general free boundaries not restricted to graphs.
@article{HL21JDE,
author ={Hao, Chengchun and Luo, Tao},
journal={Journal of Differential Equations},
title ={Well-posedness for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations},
year ={2021},
volume ={299},
number ={},
pages ={542-601},
doi={10.1016/j.jde.2021.07.030},
}
- Hao, Chengchun and Luo, Tao;
Ill-posedness of free boundary problem of the incompressible ideal MHD
Communications in Mathematical Physics, 376(1): 259–286, 2020.
In the present paper, we show the ill-posedness of the free boundary problem of the incompressible ideal magnetohydrodynamics (MHD) equations in two spatial dimensions for any positive vacuum permeability $\mu _0$, in Sobolev spaces. The analysis is uniform for any $\mu_0>0$.
@article{HL20CMP,
author ={Hao, Chengchun and Luo, Tao},
journal={Communications in Mathematical Physics},
title ={Ill-posedness of free boundary problem of the incompressible ideal MHD},
year ={2020},
volume ={376},
number ={1},
pages ={259–286},
doi={10.1007/s00220-019-03614-1},
}
- Hao, Chengchun;
On the motion of free interface in ideal incompressible MHD
Archive for Rational Mechanics and Analysis, 224(2): 515–553, 2017.
For the free boundary problem of the plasma–vacuum interface to 3D ideal incompressible magnetohydrodynamics, the a priori estimates of smooth solutions are proved in Sobolev norms by adopting a geometrical point of view and some quantities such as the second fundamental form and the velocity of the free interface are estimated. In the vacuum region, the magnetic fields are described by the div–curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic fields are tangential to the interface, but we do not need any restrictions on the size of the magnetic fields on the free interface. We introduce the “fictitious particle” endowed with a fictitious velocity field in vacuum to reformulate the problem to a fixed boundary problem under the Lagrangian coordinates. The $ L^2$-norms of any order covariant derivatives of the magnetic fields both in vacuum and on the boundaries are bounded in terms of initial data and the second fundamental forms of the free interface and the rigid wall. The estimates of the curl of the electric fields in vacuum are also obtained, which are also indispensable in elliptic estimates.
@article{Hao17ARMA,
author ={Hao, Chengchun},
journal={Archive for Rational Mechanics and Analysis},
title ={On the motion of free interface in ideal incompressible MHD},
year ={2017},
volume ={224},
number ={2},
pages ={515–553},
doi={10.1007/s00205-017-1082-7},
}
- Hao, Chengchun and Luo, Tao;
A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows
Archive for Rational Mechanics and Analysis, 212(3): 805-847, 2014.
In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid magnetohydrodynamics equations in all physical spatial dimensions $ n = 2$ and $3$ by adopting a geometrical point of view used in Christodoulou and Lindblad (Commun Pure Appl Math 53:1536–1602, 2000), and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids.
@article{HL14ARMA,
author ={Hao, Chengchun and Luo, Tao},
journal={Archive for Rational Mechanics and Analysis},
title ={A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows},
year ={2014},
volume ={212},
number ={3},
pages ={805-847},
doi={10.1007/s00205-013-0718-5},
}
♣ Two-phase equations
- Zhang, Wei and Fu, Jie and Hao, Chengchun and Yang, Siqi;
Global well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximation
Journal of Mathematical Physics, 65(8): No.081509, 22pp, 2024.
This paper focuses on the global well-posedness of the Oberbeck-Boussinesq approximation for the unsteady motion of a drop in another bounded fluid separated by a closed interface with surface tension. We assume that the initial states of the drop are close to a ball $ B_{R}$ with the same volume as the drop, and that the boundary of the drop is a small perturbation of the boundary of $ B_{R}$. To begin, we introduce the Hanzawa transformation with an added barycenter point to obtain the linearized Oberbeck-Boussinesq approximation in a fixed domain. Then, we establish time-weighted estimates of solutions for the shifted equation using maximal $ L^p$-$ L^q$ regularities for the two-phase fluid motion of the linearized system, as obtained by Hao and Zhang in 2022. Using the time decay estimates of the semigroup, we obtain decay time-weighted estimates of solutions for the linearized problem. Additionally, we prove that these estimates are less than the sum of the initial value and its own square and cube by estimating the corresponding non-linear terms. Finally, the existence and uniqueness of solutions in the finite time interval $(0,T)$ was proven by Hao and Zhang in 2023. After that, we demonstrate that the solutions can be extended beyond $ T$ by analyzing the properties of the roots of algebraic equations.
@article{ZFHY24a,
author ={Zhang, Wei and Fu, Jie and Hao, Chengchun and Yang, Siqi},
journal={Journal of Mathematical Physics},
title ={Global well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximation},
year ={2024},
volume ={65},
number ={8},
pages ={No.081509, 22pp},
doi={10.1063/5.0220764},
}
- Hao, Chengchun and Zhang, Wei;
Local well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximation
Communications on Pure and Applied Analysis, 22(7): 2099-2131, 2023.
This paper is concerned with the local well-posedness of the Oberbeck-Boussinesq approximation for the unsteady motion of a drop in another fluid separated by a closed interface with surface tension. We devote to getting the linearized Oberbeck-Boussinesq approximation in the fixed domain by using the Hanzawa transformation, and using maximal $ L^{p}$-$ L^{q}$ regularities for the two-phase fluid motion of the linearized system obtained by the authors in [11] to establish the existence and uniqueness of the solutions of nonlinear problem with the help of the contraction mapping principle, in which the differences of nonlinear terms are estimated.
@article{HZ23CPAA,
author ={Hao, Chengchun and Zhang, Wei},
journal={Communications on Pure and Applied Analysis},
title ={Local well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximation},
year ={2023},
volume ={22},
number ={7},
pages ={2099-2131},
doi={10.3934/cpaa.2023059},
}
- Hao, Chengchun and Zhang, Wei;
Maximal $ L^p$-$ L^q$ regularity for two-phase fluid motion in the linearized Oberbeck-Boussinesq approximation
Journal of Differential Equations, 322: 101-134, 2022.
This paper is concerned with the generalized resolvent estimate and the maximal $ L^p$-$ L^q$ regularity of the linearized Oberbeck-Boussinesq approximation for unsteady motion of a drop in another fluid without surface tension, which is indispensable for establishing the well-posedness of the Oberbeck-Boussinesq approximation for the two incompressible liquids separated by a closed interface. We prove the existence of $\mathcal{R}$-bounded solution operators for the model problems and the maximal $ L^p$-$ L^q$ regularity for the system. The key step is to prove the maximal $ L^p$-$ L^q$ regularity theorem for the linearized heat equation with the help of the $\mathcal{R}$-bounded solution operators for the corresponding resolvent problem and the Weis operator-valued Fourier multiplier theorem.
@article{HZ22JDE,
author ={Hao, Chengchun and Zhang, Wei},
journal={Journal of Differential Equations},
title ={Maximal $ L^p$-$ L^q$ regularity for two-phase fluid motion in the linearized Oberbeck-Boussinesq approximation},
year ={2022},
volume ={322},
number ={},
pages ={101-134},
doi={10.1016/j.jde.2022.03.022},
}
♣ Elastodynamics
- Fu Jie, Hao Chengchun*, Yang Siqi, Zhang Wei;
A blow-up criterion for free boundary problem of incompressible neo-Hookean elastodynamics
SCIENTIA SINICA Mathematica, 55: 1–16, 2025.
We prove a Beale-Kato-Majda type blow-up criterion for the solution to the free boundary problem
of the three-dimensional incompressible neo-Hookean elastodynamics model. The result indicates that, under the
Taylor-type sign condition, as long as some norms of the curl of the deformation tensor and the velocity field, the
second fundamental form of the free boundary, the injective radius of the normal exponential map, and pressure
remain bounded, and the gradients of the deformation tensor and the velocity field, together with the material
derivatives of the gradient of pressure remain bounded on the free boundary, the solution can be continuously
extended.
- Hao Chengchun*, Wang Dehua;
A priori estimates for the free boundary problem of incompressible neo-Hookean elastodynamics
Journal of Differential Equations 261(1), 712–737, 2016.
A free boundary problem for the incompressible neo-Hookean elastodynamics is studied in two and three spatial dimensions. The a priori estimates in Sobolev norms of solutions with the physical vacuum condition are established through a geometrical point of view of Christodoulou and Lindblad (2000) [3]. Some estimates on the second fundamental form and velocity of the free surface are also obtained.
♣ Euler equations
- Hao, Chengchun;
Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities
Discrete and Continuous Dynamical Systems - B, 20(9): 2885-2931, 2015.
In this paper, we establish a priori estimates for three-dimensional compressible Euler equations with the moving physical vacuum boundary, the $\gamma$-gas law equation of state for $ \gamma=2$ and the general initial density in \(H^5\). Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which plays a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $ L^3$.
@article{Hao15,
author ={Hao, Chengchun},
journal={Discrete and Continuous Dynamical Systems - B},
title ={Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities},
year ={2015},
volume ={20},
number ={9},
pages ={2885-2931},
doi={10.1016/j.jde.2016.03.025},
}
■ Well-posedness of Cauchy problems to fluid dynamics equations
- Hao, Chengchun and Li, Hai-Liang;
Well-posedness for a multidimensional viscous liquid-gas two-phase flow model
SIAM J. Math. Anal., 44(3): 1304–1332, 2012.
The Cauchy problem of a multidimensional ($ d\geqslant 2$) compressible viscous liquid-gas two-phase flow model is studied in this paper. We investigate the global existence and uniqueness of the strong solution for the initial data close to a stable equilibrium and the local-in-time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. A continuation criterion is also obtained for the local solution.
@article{HL12SIMA,
author ={Hao, Chengchun and Li, Hai-Liang},
journal={SIAM J. Math. Anal.},
title ={Well-posedness for a multidimensional viscous liquid-gas two-phase flow model},
year ={2012},
volume ={44},
number ={3},
pages ={1304–1332},
doi={10.1137/110851602},
}
- Hao, Chengchun;
Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces
Zeitschrift für angewandte Mathematik und Physik, 63(5): 825-834, 2012.
We investigate the Cauchy problem of a multidimensional chemotaxis model with initial data in critical Besov spaces. The global existence and uniqueness of the strong solution is shown for initial data close to a constant equilibrium state.
@article{Hao12ZAMP,
author ={Hao, Chengchun},
journal={Zeitschrift für angewandte Mathematik und Physik},
title ={Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces},
year ={2012},
volume ={63},
number ={5},
pages ={825-834},
doi={10.1007/s00033-012-0193-0},
}
- Lin, Yiquan and Hao, Chengchun and Li, Hai-Liang;
Global well-posedness of compressible bipolar Navier-Stokes-Poisson equations
Acta Mathematica Sinica, English Series, 28(5): 925-940, 2012.
We consider the initial value problem for multi-dimensional bipolar compressible Navier-Stokes-Poisson equations, and show the global existence and uniqueness of the strong solution in hybrid Besov spaces with the initial data close to an equilibrium state.
@article{LHL12AMSE,
author ={Lin, Yiquan and Hao, Chengchun and Li, Hai-Liang},
journal={Acta Mathematica Sinica, English Series},
title ={Global well-posedness of compressible bipolar Navier-Stokes-Poisson equations},
year ={2012},
volume ={28},
number ={5},
pages ={925-940},
doi={10.1007/s10114-011-0238-x},
}
- Hao, Chengchun;
Well-posedness for the viscous rotating shallow water equations with friction terms
J. Math. Phys., 52(2): No.023101, 12pp, 2011.
We consider the Cauchy problem for viscous rotating shallow water equations with friction terms. The global existence of the solution in some hybrid spaces is shown for the initial data close to a constant equilibrium state away from the vacuum.
@article{Hao11JMP,
author ={Hao, Chengchun},
journal={J. Math. Phys.},
title ={Well-posedness for the viscous rotating shallow water equations with friction terms},
year ={2011},
volume ={52},
number ={2},
pages ={No.023101, 12pp},
doi={10.1063/1.3553187},
}
- Hao, Chengchun;
Well-posedness to the compressible viscous magnetohydrodynamic system
Nonlinear Analysis: Real World Applications, 12(6): 2962–2972, 2011.
This paper is concerned with the Cauchy problem of the compressible viscous magnetohydrodynamic (MHD) system in whole spatial space $ \R^d$ for $ d\geqslant 3$. It is shown that the global solution exists uniquely in hybrid Besov spaces provided the initial data close to a constant equilibrium state away from the vacuum.
@article{Hao11NARWA,
author ={Hao, Chengchun},
journal={Nonlinear Analysis: Real World Applications},
title ={Well-posedness to the compressible viscous magnetohydrodynamic system},
year ={2011},
volume ={12},
number ={6},
pages ={2962–2972},
doi={10.1016/j.nonrwa.2011.04.017},
}
- Hao, Chengchun;
Cauchy problem for viscous shallow water equations with surface tension
Discrete Contin. Dyn. Syst. Ser. B, 13(3): 593-608, 2010.
We are concerned with the Cauchy problem for a viscous shallow water system with a third-order surface-tension term. The global existence and uniqueness of the solution in the space of Besov type is shown for the initial data close to a constant equilibrium state away from the vacuum by using the Friedrich's regularization and compactness arguments.
@article{,
author ={Hao, Chengchun},
journal={Discrete Contin. Dyn. Syst. Ser. B},
title ={Cauchy problem for viscous shallow water equations with surface tension},
year ={2010},
volume ={13},
number ={3},
pages ={593-608},
doi={10.3934/dcdsb.2010.13.593},
}
- Hao, Chengchun and Hsiao, Ling and Li, Hai-Liang;
Cauchy problem for viscous rotating shallow water equations
Journal of Differential Equations, 247(12): 3234–3257, 2009.
We consider the Cauchy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modeling of motions for shallow water with free surface in a rotating sub-domain Marche (2007) [19]. The global existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuum. Unlike the previous analysis about the compressible fluid model without Coriolis forces, see for instance Danchin (2000) [10], Haspot (2009) [16], the rotating effect causes a coupling between two parts of Hodge's decomposition of the velocity vector field, and additional regularity is required in order to carry out the Friedrichs' regularization and compactness arguments.
@article{HHL09JDE,
author ={Hao, Chengchun and Hsiao, Ling and Li, Hai-Liang},
journal={Journal of Differential Equations},
title ={Cauchy problem for viscous rotating shallow water equations},
year ={2009},
volume ={247},
number ={12},
pages ={3234–3257},
doi={10.1016/j.jde.2009.09.008},
}
- Hao, Chengchun and Li, Hai-Liang;
Global existence for compressible Navier–Stokes–Poisson equations in three and higher dimensions
Journal of Differential Equations, 246(12): 4791–4812, 2009.
The compressible Navier–Stokes–Poisson system is concerned in the present paper, and the global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces in three and higher dimensions.
@article{HL09JDE,
author ={Hao, Chengchun and Li, Hai-Liang},
journal={Journal of Differential Equations},
title ={Global existence for compressible Navier–Stokes–Poisson equations in three and higher dimensions},
year ={2009},
volume ={246},
number ={12},
pages ={4791–4812},
doi={10.1016/j.jde.2008.11.019},
}
■ Well-posedness of IVP/IBVP to dispersive equations
- Recent progress on some kinds of nonlinear dispersive partial differential equations (in Chinese)
Research Series for Excellent Doctoral Dissertation of Chinese Academy of Sciences vol. 2007, 1-6, Science Press, Beijing, 2008.
- Hao, Chengchun and Hsiao, Ling and Li, Hai-Liang;
Global well posedness for the Gross–Pitaevskii equation with an angular momentum rotational term
Math. Methods Appl. Sci., 31(6): 655-664, 2008.
In this paper, we establish the global well posedness of the Cauchy problem for the Gross–Pitaevskii equation with a rotational angular momentum term in the space $\Bbb{R}^2$.
@article{HHL08MMAS,
author ={Hao, Chengchun and Hsiao, Ling and Li, Hai-Liang},
journal={Math. Methods Appl. Sci.},
title ={Global well posedness for the Gross–Pitaevskii equation with an angular momentum rotational term},
year ={2008},
volume ={31},
number ={6},
pages ={655-664},
doi={10.1002/mma.931},
}
- Hao, Chengchun and Hsiao, Ling and Li, Hai-Liang;
Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions
J. Math. Phys., 48(10): No.102105, 11pp, 2007.
In this paper, we establish the global well posedness of the Cauchy problem for the Gross-Pitaevskii equation with an angular momentum rotational term in which the angular velocity is equal to the isotropic trapping frequency in the space $\Bbb{R}^3$.
@article{HHL07JMP,
author ={Hao, Chengchun and Hsiao, Ling and Li, Hai-Liang},
journal={J. Math. Phys.},
title ={Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions},
year ={2007},
volume ={48},
number ={10},
pages ={No.102105, 11pp},
doi={10.1063/1.2795218},
}
- Hao, Chengchun;
Well-posedness for one-dimensional derivative nonlinear Schr\"odinger equations
Commun. Pure Appl. Anal., 6(4): 997-1021, 2007.
In this paper, we investigate the one-dimensional derivative nonlinear Schr\"odinger equations of the form $ iu_t-u_{x x}+i\lambda |u|^k u_x=0$ with non-zero $\lambda\in \mathbb R$ and any real number $ k\geq 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $ H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.
@article{Hao07CPAA,
author ={Hao, Chengchun},
journal={Commun. Pure Appl. Anal.},
title ={Well-posedness for one-dimensional derivative nonlinear Schr\"odinger equations},
year ={2007},
volume ={6},
number ={4},
pages ={997-1021},
doi={10.3934/cpaa.2007.6.997},
}
- Hao, Chengchun and Hsiao, Ling and Wang, Baoxiang;
Well-posedness of Cauchy problem for the fourth order nonlinear Schr\"odinger equations in multi-dimensional spaces
Journal of Mathematical Analysis and Applications, 328(1): 58-83, 2007.
@article{HHW07JMAA,
author ={Hao, Chengchun and Hsiao, Ling and Wang, Baoxiang},
journal={Journal of Mathematical Analysis and Applications},
title ={Well-posedness of Cauchy problem for the fourth order nonlinear Schr\"odinger equations in multi-dimensional spaces},
year ={2007},
volume ={328},
number ={1},
pages ={58-83},
doi={10.1016/j.jmaa.2006.05.031},
}
- Hao, Chengchun and Hsiao, Ling and Wang, Baoxiang;
Wellposedness for the fourth order nonlinear Schr\"odinger equations
Journal of Mathematical Analysis and Applications, 320(1): 246-265, 2006.
We study the local smoothing effects and wellposedness of Cauchy problem for the fourth order nonlinear Schr\"odinger equations in 1D
$$
i\partial_t u=\partial_x^4 u+P\left((\partial_x^\alpha u)_{\abs{\alpha}\leqslant 2}, (\partial_x^\alpha
\bar{u})_{|\alpha|\leqslant 2}
ight),\quad t,x\in \Bbb{R},
$$
where $ P(\cdot)$ is a polynomial excluding constant and linear terms.
@article{HHW06JMAA,
author ={Hao, Chengchun and Hsiao, Ling and Wang, Baoxiang},
journal={Journal of Mathematical Analysis and Applications},
title ={Wellposedness for the fourth order nonlinear Schr\"odinger equations},
year ={2006},
volume ={320},
number ={1},
pages ={246-265},
doi={10.1016/j.jmaa.2005.06.091},
}
- Wang, Baoxiang and Hao, Chengchun and Hudzik, Henryk;
Energy scattering theory for the nonlinear Schr\"odinger equations with exponential growth in lower spatial dimensions
Journal of Differential Equations, 228(1): 311-338, 2006.
For one and two spatial dimensions, we show the existence of the scattering operators for the nonlinear Schr\"odinger equation with exponential nonlinearity in the whole energy spaces.
@article{WHH06JDE,
author ={Wang, Baoxiang and Hao, Chengchun and Hudzik, Henryk},
journal={Journal of Differential Equations},
title ={Energy scattering theory for the nonlinear Schr\"odinger equations with exponential growth in lower spatial dimensions},
year ={2006},
volume ={228},
number ={1},
pages ={311-338},
doi={10.1016/j.jde.2006.05.010},
}
- Hao, Chengchun;
The initial boundary value problem for quasi-linear Schr\"odinger-Poisson equations
Acta Math. Sci. Ser. B Engl. Ed., 26(1): 115-124, 2006.
In this article, the author studies the initial-(Dirichlet) boundary problem for a high-field version of the Schr\"odinger-Poisson equations, which include a nonlinear term in the Poisson equation corresponding to a field-dependent dielectric constant and an effective potential in the Schr\"odinger equations on the unit cube. A global existence and uniqueness is established for a solution to this problem.
@article{Hao06AMSBE,
author ={Hao, Chengchun},
journal={Acta Math. Sci. Ser. B Engl. Ed.},
title ={The initial boundary value problem for quasi-linear Schr\"odinger-Poisson equations},
year ={2006},
volume ={26},
number ={1},
pages ={115-124},
doi={10.1016/S0252-9602(06)60033-7},
}
- Hao, Chengchun and Hsiao, Ling;
Studies on Schr\"odinger-Poisson systems (Excerpt of Dissertation)
J. Grad. Sch. of CAS, 22(5): 141-146, 2005.
The bipolar(defocusing nonlinear)Schr\"odinger-Poisson system and quasi—linear Schr\"odinger—Poisson equations ale studied．The wellposedness，large time behavior and modified scattering theory is established for the Cauchy problem to the bipolar(defocusing nonlinear)Schr\"odinger—Poisson systems．The initial-(Dirichlet) boundary problem for a high field version of the Schr\"odinger-Poisson equations，quasi-linear Schr\"odinger-Poisson equations，which include a nonlinear term in the Poisson equation
corresponding to a field-dependent dielectric constant and an effective potential in the Schr\"odinger equations on the unit cube ale also discussed．
@article{HH05JCAS,
author ={Hao, Chengchun and Hsiao, Ling},
journal={J. Grad. Sch. of CAS},
title ={Studies on Schr\"odinger-Poisson systems (Excerpt of Dissertation)},
year ={2005},
volume ={22},
number ={5},
pages ={141-146},
}
- Hao, Chengchun and Li, Hai-Liang;
On the initial value problem for the bipolar Schr\"odinger-Poisson systems
J. Partial Diff. Eqns., 17(3): 283-288, 2004.
In this paper，we prove the existence and uniqueness of global solutions in $ H^s(\Bbb{R}^3)$ ($ s\in \Bbb{R}，s\geqslant 0$)for the initial value problem of the bipolar Schr\"odinger-Poisson systems.
@article{HL04JPDE,
author ={Hao, Chengchun and Li, Hai-Liang},
journal={J. Partial Diff. Eqns.},
title ={On the initial value problem for the bipolar Schr\"odinger-Poisson systems},
year ={2004},
volume ={17},
number ={3},
pages ={283-288},
}
- Hao, Chengchun and Hsiao, Ling and Li, Hai-Liang;
Modified scattering for bipolar nonlinear Schr\"odinger-Poisson equations
Math. Models Methods Appl. Sci., 14(10): 1481-1494, 2004.
In this paper, we study the asymptotic behavior in time and the existence of the modified scattering operator of the globally defined smooth solutions to the Cauchy problem for the bipolar nonlinear Schr\"odinger–Poisson equations with small data in the space $ \R^3$.
@article{HHL04M3AS,
author ={Hao, Chengchun and Hsiao, Ling and Li, Hai-Liang},
journal={Math. Models Methods Appl. Sci.},
title ={Modified scattering for bipolar nonlinear Schr\"odinger-Poisson equations},
year ={2004},
volume ={14},
number ={10},
pages ={1481-1494},
doi={10.1142/S0218202504003684},
}
- Hao, Chengchun and Hsiao, Ling;
Large time behavior and global existence of solution to the bipolar defocusing nonlinear Schr\"odinger-Poisson system
Quart. Appl. Math., 62(4): 701-710, 2004.
In this paper, we study the large time behavior and the existence of globally defined smooth solutions to the Cauchy problem for the bipolar defocusing nonlinear Schr\"odinger-Poisson system in the space $\R^3$.
@article{HH04QAM,
author ={Hao, Chengchun and Hsiao, Ling},
journal={Quart. Appl. Math.},
title ={Large time behavior and global existence of solution to the bipolar defocusing nonlinear Schr\"odinger-Poisson system},
year ={2004},
volume ={62},
number ={4},
pages ={701-710},
doi={10.1090/qam/2104270},
}
- Hao, Chengchun;
Energy scattering for the generalized Davey-Stewartson equations
Acta. Math. Appl. Sinica, 19(2): 333-340, 2003.
@article{Hao03AMAS,
author ={Hao, Chengchun},
journal={Acta. Math. Appl. Sinica},
title ={Energy scattering for the generalized Davey-Stewartson equations},
year ={2003},
volume ={19},
number ={2},
pages ={333-340},
doi={10.1007/s10255-003-0108-0},
}
- Energy scattering for nonlinear Davey-Stewartson equations (in Chinese)
J. Math. Res. Expo. 23(4), 645-650, 2003.
In this paper, the author studies the scattering theory of a class of nonlinear Davey-Stewartson equations with three nonlinearities. The author proves that their scattering operator exists in the energy space $H^1$.
- Energy scattering for the generalized Davey-Stewartson equations (Rapid Communications)
Hebei Uni., Rat. Sci. Edt. 22(1), 73-74, 2002.
■ Well-posedness or asymptotic of IVP/IBVP to quantum models
- Li, Hai-Liang and Zhang, Guo-Jing and Zhang, Min and Hao, Chengchun;
Long-time self-similar asymptotic of the macroscopic quantum models
J. Math. Phys., 49(7): No.073503, 14pp, 2008.
The unipolar and bipolar macroscopic quantum models derived recently, for instance, in the area of charge transport are considered in spatial one-dimensional whole space in the present paper. These models consist of nonlinear fourth-order parabolic equation for unipolar case or coupled nonlinear fourth-order parabolic system for bipolar case. We show for the first time the self-similarity property of the macroscopic quantum models in large time. Namely, we show that there exists a unique global strong solution with strictly positive density to the initial value problem of the macroscopic quantum models which tends to a self-similar wave (which is not the exact solution of the models) in large time at an algebraic time-decay rate.
@article{LZZH08JMP,
author ={Li, Hai-Liang and Zhang, Guo-Jing and Zhang, Min and Hao, Chengchun},
journal={J. Math. Phys.},
title ={Long-time self-similar asymptotic of the macroscopic quantum models},
year ={2008},
volume ={49},
number ={7},
pages ={No.073503, 14pp},
doi={10.1063/1.2949082},
}
- Hao, Chengchun and Jia, Yueling and Li, Hai-Liang;
Quantum Euler-Poisson system: local existence of solutions
J. Partial Diff. Eqns., 16(4): 306-320, 2003.
The one-dim ensional tran sient quantum Euler-Poisson system for sem i-conductors iS studied in a bounded interva1．The quan tum correction can be interpreted as a dispersive regularization of the classical hydrodynamic equations and mechanical efects．
The existence an d uniqueness of local-in-time solutions are proved with lower regularity an d without the restriction on the smallness of velocity，where the pressure—density is general(can be non—convex or non-monotone)．
@article{HJL03JPDE,
author ={Hao, Chengchun and Jia, Yueling and Li, Hai-Liang},
journal={J. Partial Diff. Eqns.},
title ={Quantum Euler-Poisson system: local existence of solutions},
year ={2003},
volume ={16},
number ={4},
pages ={306-320},
}
Published Books
已出版著作
Wang, Baoxiang and Huo, Zhaohui and Hao, Chengchun and Guo, Zihua;
Harmonic Analysis Method for Nonlinear Evolution Equations,I
World Scientific Pub Co Inc 300pp, 2011.
[中文版: 非线性发展方程的调和分析方法(I)适定性理论 (未出版, 2010) 下载]
This monograph provides a comprehensive overview on a class of nonlinear evolution equations, such as nonlinear Schrödinger equations, nonlinear Klein–Gordon equations, KdV equations as well as Navier–Stokes equations and Boltzmann equations. The global wellposedness to the Cauchy problem for those equations is systematically studied by using the harmonic analysis methods.
This book is self-contained and may also be used as an advanced textbook by graduate students in analysis and PDE subjects and even ambitious undergraduate students.
2025-01-20
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